Linear and Nonlinear Fokker-Planck Equations

Stochastic variational principles for the collisional Vlasov–Maxwell and Vlasov–Poisson equations

“…Therefore, in this work we will be interested only in those collision operators for which (1.1) takes the form of a linear or strongly nonlinear Fokker–Planck equation (e.g. [52–54]). Namely, we will assume that the collision operator can be expressed as Cfalse[ffalse]=12∑i,j=13normal∂2normal∂vinormal∂vj[Dijfalse(boldx,boldv;ffalse)f]−∑i=13∂normal∂vi[Kifalse(boldx,boldv;ffalse)f], for some symmetric positive semi-definite matrix Dijfalse(boldx,boldv;ffalse) and vector Kifalse(boldx,boldv;ffalse) functions, where the dependence of Dij and Ki on f may in general be nonlinear, and may involve differential and integral forms of f .…”

“…Namely, we will assume that the collision operator can be expressed as Cfalse[ffalse]=12∑i,j=13normal∂2normal∂vinormal∂vj[Dijfalse(boldx,boldv;ffalse)f]−∑i=13∂normal∂vi[Kifalse(boldx,boldv;ffalse)f], for some symmetric positive semi-definite matrix Dijfalse(boldx,boldv;ffalse) and vector Kifalse(boldx,boldv;ffalse) functions, where the dependence of Dij and Ki on f may in general be nonlinear, and may involve differential and integral forms of f . In that case (1.1) is an integro-differential equation, the so-called strongly nonlinear Fokker–Planck equation [52]. In case Dij and …”

“…If (1.1) has the form of a Fokker–Planck equation, then the particle density function f can be interpreted as the probability density function for a stochastic process false(boldXfalse(tfalse),boldVfalse(tfalse)false)∈double-struckR3×double-struckR3. This stochastic process then satisfies the Stratonovich stochastic differential equation [52–55] dboldX=boldV dt and dboldV=(qmboldEfalse(boldX,tfalse)+qmboldV×boldBfalse(boldX,tfalse)+boldGfalse(boldX,boldV;ffalse)) dt+∑ν=1Mgνfalse(boldX,boldV;ffalse)∘dWνfalse(tfalse), where W1false(tfalse),…,W<...>…”

Brownian Swarm Dynamics and Burgers’ Equation with Higher Order Dispersion

“…Equation ( 6) describes agents mutually interacting via their own repartition density. The homogeneous character of the interactions of Equation ( 6) and the specific choice of WGN stochastic driving enable one to formally write the collective evolution by means of a nonlinear parabolic PDE (i.e., a nonlinear Fokker-Planck (FP) equation) [16]:…”

“…Such interactions can be viewed as special cases (i.e., limited to large swarms populations) of the more general class of dynamics introduced in [14,15]. In the specific context of Brownian agents where the evolution is Markovian, Equation (1) with specific boundary conditions can be alternatively interpreted as a nonlinear Fokker-Planck equation [16]. With this probabilistic interpretation, kink solutions directly describe travelling probability distributions of the form P(ξ − vt) := P(x) with ξ ∈ R and v the travelling velocity.…”